Error of Analysis of Newton-Cotes formulas

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Error Analysis of Newton-Cotes formulas[edit]

The Newton-Cotes formulas are a group of formulas for evaluating numeric integration at equally spaced points.

The Methods[1][edit]

Let , , be equally spaced points, and be the corresponding values. Let be the space , and let be the interpolation variable . Thus to interpolate at x,

A polynomial of degree can be derived to pass through these points and approximate the function . Using divided differences and Newton polynomial, can be obtained as

From the general form of polynomial interpolation error, the error of using to interpolate can be obtained as

where .

Since , the error term of numerical integration is

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(1)

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Error terms for different rules[edit]

The Trapezoid Rule[edit]

Let's consider the trapezoid rule in a single interval. In each interval, the integration uses two end points. Thus . Then . Applying (1 ), we get

where . Thus the local error is . Consider the composite trapezoid rule. Given that , the global error is

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(2)

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where , .

To justify (2 ), we can need the theorem below[2] in page 345:

If  is continuous and the , then for some value  in the interval of all the arguments 

The Simpson's 1/3 Rule[edit]

Consider Simpson's 1/3 rule. In this case, three equally spaced points are used for integration. Thus . Applying (1 ), we get

where .

This doesn't mean that the error is zero. It simply means that the cubic term is identically zero. The error term can be obtained from the next term in the Newton polynomial, obtaining

Thus the local error is and the global error is .

The Simpson's 3/8 Rule[edit]

Consider Simpson's 3/8 rule. In this case, since four equally spaced points are used. Applying (1 ), we get

where .

Both the Simpon's 1/3 rule and the 3/8 rule have error terms of order . With smaller coefficient, the 1/3 rule seems more accurate. Then why do we need the 3/8 rule? The 3/8 rule is useful when the total number of increments is odd. Three increments can be used with the 3/8 rule, and then the rest even number of increments can be used with 1/3 rule.

A Numerical Example[edit]

Given the set of data points, solve the numerical integration

3.1 -0.32258065
3.5 -0.28571429
3.9 -0.25641026

Solution[edit]

Use the trapezoid rule. First try . That is, use only the two end points. We can get

Compared with the exact solution we have

Using all three points with we can get

and so

Thus the error ratio is . This is close to what we can get by inspecting

.

Exercises[edit]

Exercise 1[3][edit]

Using the data given below, find the maximum error incurred in using Newton's forward interpolation formula to approximate .

0.1 1.10517
0.2 1.22140
0.3 1.34986
0.4 1.49182
0.5 1.64872

Exercise 2[edit]

When using Simpson's 1/3, what is the error ratio supposed to be?

References[edit]

  1. Hoffman, Joe D. (2001). Numerical Methods for Engineers and Scientists (2nd ed.). Marcel Derkker, INC. ISBN 0-8247-0443-6.
  2. Hamming, R. W. (1986). Numerical Methods for Scientists and Engineers (2nd ed.). New York: Dover Publications. ISBN 0-486-65241-6.
  3. Tenenbaum, Morris; Pollard, Harry (1985). Ordinary Differential Equations: An Elementary Textbook for Students of Mathematics, Engineering, and the Sciences. New York: Dover Publications. ISBN 0-786-64940-7 Check |isbn= value: checksum (help).CS1 maint: Multiple names: authors list (link)